implying Xk∈I and consequently (Xk)⊆I.
For obtaining the reverse inclusion, suppose that

h⁢(X):=bn⁢Xn+bn+1⁢Xn+1+…

is an arbitrary nonzero element of I where bn≠0. Because n≥k, we may write

h⁢(X)=Xk⁢(bn⁢Xn-k+bn+1⁢Xn-k+1+…).

This equation says that h⁢(X)∈(Xk), whence I⊆(Xk).
Thus we have seen that I is the principal ideal (Xk), so that K⁢[[X]] is a principal ideal domain.
Now, all ideals of the ring K⁢[[X]] form apparently the strictly descending chain

(X)⊃(X2)⊃(X3)⊃…⊃(0),

whence the ring has the unique maximal ideal (X). A principal ideal domain with only one maximal ideal is a discrete valuation ring.